(Revised Core Set Perspective)
I have tried to make this card work time and time again in tagstorm NBN: Making News but I think it's safe to say NBN should stick to always watching rather than responding.
Dedicated Response Team is a thematically awesome card that goes along well with Weyland's strategy of kicking down doors, burning down buildings, and sending in the Cleaners afterwards. It's often compared to the remaining tag punishment cards in the RCS: Private Security Force, Closed Accounts, and Psychographics. Score a Cleaners quickly and this can really gain some massive firepower with 3 damage per successful run.
The problem is just that though; for DRT to fire, it requires the runner to make a successful run. Compared to the other 3 tag punishment cards above, each can be used immediately after landing a tag on the runner with a SEA Source. As such, these punishment cards can be used at the corp's discretion.
As an asset, DRT suffers from having to be on standby. It requires tag support in the form of cards that will tag mid-run - any other time and the runner will remove the tag on their turn. Currently, Weyland does not have the support cards necessary as Shadow's subs are often broken due to its low strength and Hunter is often overlooked compared to other Weyland sentries like Archer or Caduceus which both come with ETR subs.
This card might find a home in NBN decks with plenty of tag-friendly cards like Data Raven, TGTBT, or Ghost Branch if it weren't for its high influence cost of 3. An NBN player has to make a decision whether to include a few of these cards for consistent flatlines (as 1 DRT will very rarely secure the flatline alone) or save the influence altogether. Not to mention having to protect the DRTs constantly from being trashed by support cards like Scrubber or Imp or a rich runner in general.
All in all, I feel a simple change to this card could have made it more consistent for corps to use (Weyland or NBN): Adding a "This applies even during the run on which the runner trashes Dedicated Response Team." like Strongbox or Red Herrings. But perhaps that's asking for too much.